Core Parties

In spatial voting games, the Condorcet winner is any policy or candidate that defeats all other [Page 148]policies or candidates in pairwise votes. However, what can we say about potential outcomes if there is no Condorcet winner? Let us define a set X as the set of feasible outcomes corresponding to vectors that denote the utility u of each player (ui, u2, u3,) to be a specific vector (i.e., point) in X. Assume that a coalition C prefers u to u′ if and only if ui > u,I for all individuals I in C. Define the coalition of individuals C to be effective for u in X if the members of C can coordinate their actions sufficiently to ensure that each member i of C receives a payoff of at least ui. Let v(C) denote the set of all utility n-tuples for which C is effective. Now we say that u dominates u′ if there exists at least one coalition that is effective for u and that prefers u to u′. A cooperative game's core is then defined as the set of undominated elements of X.

A core party is then defined as any party that in a voting game occupies the core of the game. In coalition theory, core parties will always form the government or be an element in a coalition government. Formal coalition theory has demonstrated that majority voting rules always generate a core in one-dimensional policy space, which means that there will always be a core party where politics is dominated by one dimension—such as a left-right ideological dimension. However, if we assume that the preferences of players are randomly assigned, then a core is generated only unusually in two dimensions and rarely if ever in three or more dimensions. It is usually assumed that under such circumstances voting cycles will be generated. However, if the public's preferences are not randomly generated, but rather are socially determined to bunch together in policy space, then a core is more likely to form, and so will core parties. Empirically, in many systems, some parties are usually members of coalition governments, and such parties are core parties within those systems.